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Extra info for Advanced Robust and Nonparametric Methods in Efficiency Analysis: Methodology and Applications (Studies in Productivity and Efficiency)

Example text

The isoquants or efficient boundaries of the sections of Ψ can be defined in radial terms (Farrell, 1957) as follows. 6) and in the output space: ∂P (x) = {y|y ∈ P (x), λy ∈ P (x), ∀λ > 1}. 7) The axiomatic approach to production theory (Activity Analysis framework) assumes that the technology (production model) satisfies certain properties or axioms. These properties can be equivalently stated on Ψ, P (x), x ∈ Rp+ , C(y), y ∈ Rq+ . Some economic axioms (EA) are usually done in this framework (on these concepts see also Shephard, 1970).

Nevertheless, it can also be used to construct input quantity indexes (Tornqvist, 1936; Malmquist, 1953) and productivity indexes (Caves, Christensen, and Diewert, 1982). Similarly, the direct output distance function introduced by Shephard (1970) and the two indirect distance functions of Shephard (1974) can be used to characterize the structure of efficient production 1 This section is based on F¨ are, Grosskopf and Lovell (1994), pp. 1-23; and Kumbhakar and Lovell (2000), pp. 5-7. A short history of thought 17 technology in the multi-product case, to measure efficiency to that technology, and to construct output quantity indexes (Bergson,1961; Moorsteen, 1961) and productivity indexes (F¨are, Grosskopf, and Lovell, 1992).

10) and its Farrell output measure of efficiency is defined as: λ(x0 , y0 ) = sup{λ|λy0 ∈ P (x0 )} = sup{λ|(x0 , λy0 ) ∈ Ψ}. 11) So, θ(x0 , y0 ) ≤ 1 is the radial contraction of inputs the firm should achieve to be considered as being input-efficient in the sense that (θ(x0 , y0 )x0 , y0 ) is a frontier point. In the same way λ(x0 , y0 ) ≥ 1 is the proportionate increase of output the firm should achieve to be considered as being output efficient in the sense that (x0 , λ(x0 , y0 )y0 ) is on the frontier.